Some generalizations of the Hardy-Littlewood theorem
A series \(\sum^\infty_{n=1} u_n\) is said to be summable to the value \(s\) by a \(P =(p_{n,k})\) method (we write \(\sum^\infty_{n=0} u_n= s(p_{n,k}))\) if \[ \lim_{n\to\infty} \sum_{k=0}^\infty p_{n,k} U_k=s \quad \text{where} \quad U_k= \sum^k_{i=1} u_i. \] The author proves the following result which is an analogue of the Hardy-Littlewood theorem on summation of the Cauchy product of two series for Cesàro summability. Theorem. Let \(c_{n,k} =-1\) for \(k=n\), 2 for \(k=n-1\), and 0 for \(k\neq n\), \(k\neq n-1\), \((n=0,1,2, \dots)\). If the series \(\sum^\infty_{n=0} u_n\) is absolutely convergent to a value \(U\) and \[ \sum^\infty_{n=1} v_n= V(c_{n,k}), \quad \text{then} \quad \sum^\infty_{n=0} w_n= UV(c_{n, k}) \quad \text{when} \quad w_n= \sum^\infty_{k=0} u_kv_{n-k}. \]
scientific article; zbMATH DE number 1028786